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Last Fall Degree, HFE, and Weil Descent Attacks on ECDLP

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Last Fall Degree, HFE, and Weil Descent Attacks on ECDLP Last fall degree, HFE, and Weil descent attacks on ECDLP Ming-Deh A. Huang (USC, [email protected]), Michiel Kosters (TL@NTU, [email protected]), Sze Ling Yeo (I2R, [email protected]) Abstract. Weil descent methods have recently been applied to attack the Hidden Field Equation (HFE) public key systems and solve the el- liptic curve discrete logarithm problem (ECDLP) in small characteristic. However the claims of quasi-polynomial time attacks on the HFE sys- tems and the subexponential time algorithm for the ECDLP depend on various heuristic assumptions. In this paper we introduce the notion of the last fall degree of a poly- nomial system, which is independent of choice of a monomial order. We then develop complexity bounds on solving polynomial systems based on this last fall degree. We prove that HFE systems have a small last fall degree, by showing that one can do division with remainder after Weil descent. This allows us to solve HFE systems unconditionally in polynomial time if the degree of the defining polynomial and the cardinality of the base field are fixed. For the ECDLP over a finite field of characteristic 2, we provide com- putational evidence that raises doubt on the validity of the first fall de- gree assumption, which was widely adopted in earlier works and which promises sub-exponential algorithms for ECDLP. In addition, we con- struct a Weil descent system from a set of summation polynomials in which the first fall degree assumption is unlikely to hold. These exam- ples suggest that greater care needs to be exercised when applying this heuristic assumption to arrive at complexity estimates. These results taken together underscore the importance of rigorously bounding last fall degrees of Weil descent systems, which remains an in- teresting but challenging open problem. Keywords: HFE · ECDLP · Weil descent · solving equations · first fall degree · last fall degree 1 Introduction 1.1 Zero-dimensional polynomial systems and Weil descent attacks Zero-dimensional multivariate polynomial systems over finite fields arise in many practical areas of interest including in cryptography and coding theory. As such, solving these systems has both practical and theoretical interest. For instance, a major application is in the design of multivariate cryptosystems. One of the earliest proposals for the multivariate cryptosystem was the HFE public-key cryptosystem [21]. In recent years, polynomial solving also arises in elliptic curve cryptography, specifically in the index calculus approach to solve the elliptic curve discrete logarithm problem (ECDLP). Many different approaches had been proposed to solve multivariate polyno- mial equations over finite fields. The most common approach for generic polyno- mial systems is via Gr¨obnerbasis algorithms [1,9,10]. Typically, a Gr¨obnerbasis with respect to the degree reverse lexicographical ordering is first computed via algorithms F4 or F5 [9,10]. It is then converted to a Gr¨obnerbasis with respect to the lexicographical ordering by algorithms such as the FGLM algorithm [11] which contains equations where variables are eliminated. This enables the vari- ables to be solved one at a time. In general, it is very difficult to determine the complexity of the Gr¨obnerbasis algorithm. Various authors have used the term \the degree of regularity" to describe properties of a system that can be used to obtain complexity results. However not all definitions of this term are equivalent. Another approach to solve multivariate polynomial systems is the XL algo- rithm and its variants [2{5,17]. This class of algorithms performs well when the system under consideration is overdetermined, that is, the number of equations far exceeds the number of variables. In this present paper, we first introduce the notion of the last fall degree of a polynomial system over a finite field. Our definition is intrinsic to the polynomial system itself, independent of the choice of a monomial order. With this notion at our disposal, we present an explicit algorithm to find all the roots of a zero- dimensional multivariate polynomial system, bounding the complexity by the last fall degree. When the polynomial systems are over a field of cardinality qn; where q is a prime power and n a positive integer, one can convert this system via Weil descent to a system over its subfield with q elements (see Section 3 for more details). This results in a polynomial system over a smaller field, but at the expense of more variables. For example, Weil descent has been adopted to solve the HFE system as well as the index calculus method for ECDLP. In this paper, we will describe Weil descent systems arising from a polynomial in one variable and study the relations among various polynomial systems. Analogous definitions hold for a multivariate polynomial system. 1.2 The HFE cryptosystem Let k be a finite field of cardinality qn, with subfield k0 of cardinality q. Let f 2 k[X] be a polynomial over k with a relatively small degree. Using factorization algorithms, one can easily factorize this polynomial to find its roots in k. One can transform this system using Weil descent and two transformations into a system in n variables over k0. At first glance, this system seems to be hard to solve and this is the basis of the Hidden Field Equations (HFE) cryptosystem (see [21] and Subsection 4.1). Computational and heuristic evidence show that such a system is not secure [7,8,16]: the degree of regularity of a Weil descent systemis small and does not depend on n and hence the system can be solved efficiently using Gr¨obnerbasis algorithms. In particular in [16], the authors claimed that the HFE system can be solved efficiently under a heuristic assumption on the 2 complexity of the Gr¨obnerbasis computations to solve the system. More recently, Christophe Petit, in a preprint [22], gives a proof of this observation by doing manipulations using his successive resultant algorithm on the descent side. In this article, we prove that HFE systems have a small last fall degree, by showing that one can do division with remainder after Weil descent. This allows us to solve the HFE systems unconditionally in polynomial time if the degree of the defining polynomial and the cardinality of the base field are fixed. 0 0n 0 We have a natural right action of Affn(k ) = k o GLn(k ) on the ring 0 0 0 R = k [Y0;:::;Yn−1] by acting as affine change of coordinates. If M 2 Affn(k ) and g 2 R0 we write gM for this action. The main theorem is the following, which allows one to solve HFE systems efficiently. We stress that our results hold for a larger class of polynomial systems as we do not require the resulting Weil descent system to be quadratic. For r 2 Z≥0 and c 2 Z≥1; we set (r; c) = max (b2(c − 1) (logc (r) + 1)c; 0) : Main Theorem 1. Let q be a prime power and let k be a finite field of cardi- nality qn with subfield k0 of cardinality q. Let f 2 k[X] nonzero which has at 0 0 most e different roots over k and let F = ffg. Let Ff ⊂ k [Y0;:::;Yn−1] be a 0 0 Weil descent system of F (Subsection 3.1). Let M 2 Affn(k ), N 2 GLn(k ). Define gi, i = 0; : : : ; n − 1, by 0 1 0 1 g0 [f]0M B . C B . C @ . A = N @ . A : gn−1 [f]n−1M q q Set d = max( (deg(f); q); q; e). Then given G = fg0; : : : ; gn−1;Y0 −Y0;:::;Yn−1− 0 Yn−1g ⊂ k [Y0;:::;Yn−1], one can deterministically find all solutions to G in time polynomial in (n + d)d. If one fixes q and deg(f), then the complexity to solve systems in Main Theorem 1 is polynomial in n. Note that e ≤ deg(f), but usually it is much smaller. Furthermore, in practical applications, one wants e to be small, say bounded by a constant: in this case one can solve the above system in quasi- polynomial time if q is fixed and deg(f) grows like nα. It is an open question whether variants of HFE, such as HFEv-, can be attacked by our approach. 1.3 Polynomial systems from ECDLP A major application of solving multivariate polynomial equations over a finite field k of cardinality qn is in the relation search step of the index calculus algo- 2 rithm for elliptic curves over the field [6,14,23]. Indeed, let E : y + a1xy + a3 = 3 2 x +a2x +a4x+a6, where a1; a2; a3; a4; a6 2 k, be an elliptic curve defined over k. Let P be a point on E and let Q be a point in the cyclic group generated by 3 P . The elliptic curve discrete logarithm problem seeks for an integer a such that Q = aP . The most important step in the index calculus approach is to generate suf- ficiently many relations among suitable points on the elliptic curve E. To this end, summation polynomials provide a way to achieve this (see Subsection 5.1). In particular, this transforms the problem of finding relations among points to solving a system of polynomial equations over k via the summation polynomials. Cryptographic applications of Weil descent were first suggested by Frey [12], and Weil descent attacks were initially applied to elliptic curves of composite degrees over F2 [12, 15]. In [6] and [14], Weil descent was exploited to solve the ECDLP by applying the Weil descent to the summation polynomials over k. In [6], for instance, sub-exponential time estimates via this Weil descent approach were obtained for certain classes of q and n.
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